next up previous
Next: Propositions Expressing Quantification Up: FIRST ORDER THEORIES OF Previous: Replacing Modal Operators by

More Philosophical Examples-Mostly Well Known

Some sentences that recur as examples in the philosophical literature will be expressed in our notation so the treatments can be compared.

First we have ``The number of planets = 9'' and ``Necessarily 9 = 9'' from which one doesn't want to deduce ``Necessarily the number of planets = 9''. This example is discussed by Quine (1961) and (Kaplan 1969). Consider the sentences




Both are true. (56) asserts that it is not necessary that the number of planets be 9, and (57) asserts that the number of planets, once determined, is a number that is necessarily equal to 9. It is a major virtue of our formalism that both meanings can be expressed and are readily distinguished. Substitutivity of equals holds in the logic but causes no trouble, because ``The number of planets = 9'' may be written


or, using concepts, as


and ``Necessarily 9=9'' is written


and these don't yield the unwanted conclusion.

Ryle used the sentences ``Baldwin is a statesman'' and ``Pickwick is a fiction'' to illustrate that parallel sentence construction does not always give parallel sense. The first can be rendered in four ways, namely tex2html_wrap_inline1271 or tex2html_wrap_inline1273 or tex2html_wrap_inline1275 or tex2html_wrap_inline1277 where the last asserts that the concept of Baldwin is one of a statesman. The second can be rendered only as as tex2html_wrap_inline1279 or tex2html_wrap_inline1281 .

Quine (1961) considers illegitimate the sentence


obtained from ``Philip is unaware that Tully denounced Catiline'' by existential generalization. In the example, we are also supposing the truth of ``Philip is aware that Cicero denounced Catiline''. These sentences are related to (perhaps even explicated by) several sentences in our system. Tully and Cicero are taken as distinct concepts. The person is called tully or cicero in our language, and we have





We can discuss Philip's concept of the person Tully by introducing a function Concept2(p1,p2) giving for some persons p1 and p2, p1's concept of p2. Such a function need not be unique or always defined, but in the present case, some of our information may be conveniently expressed by


asserting that Philip's concept of the person Tully is Cicero. The basic assumptions of Quine's example also include




gif From (63), tex2html_wrap_inline1305 , (67) we can deduce


from (67) and others, and




using the additional hypotheses





Presumably (68) is always true, because we can always construct a concept whose denotation is Cicero unbeknownst to Philip. The truth of (69) depends on Philip's knowing that someone denounced Catiline and on the map Concept2(p1,p2) that gives one person's concept of another. If we refrain from using a silly map that gives something like Denouncer(Catiline) as its value, we can get results that correspond to intuition.

The following sentence attributed to Russell is is discussed by Kaplan: ``I thought that your yacht was longer than it is''. We can write it



where we are not analyzing the pronouns or the tense, but are using denot to get the actual length of the yacht and Concept1 to get back a concept of this true length so as to end up with a proposition that the length of the yacht is greater than that number. This looks problematical, but if it is consistent, it is probably useful.

In order to express ``Your yacht is longer than Peter thinks it is.'', we need the expression Denot(Peter,X) giving a concept of what Peter thinks the value of X is. We now write


but I am not certain this is a correct translation.

Quine (1956) discusses an example in which Ralph sees Bernard J. Ortcutt skulking about and concludes that he is a spy, and also sees him on the beach, but doesn't recognize him as the same person. The facts can be expresed in our formalism by equations




where P1 and P2 are concepts satisfying denot P1 = ortcutt and denot P2 = ortcutt. P1 and P2 are further described by sentences relating them to the circumstances under which Ralph formed them.

We can still consider a simple sentence involving the persons as things--write it believespy(ralph,ortcutt), where we define


using suitable mappings Concept1 and Concept7 from persons to concepts of persons. We might also choose to define believespy in such a way that it requires tex2html_wrap_inline1339 for several concepts P of p2, e.g. the concepts arising from all p1's encounters with p2 or his name. In this case


will be false and so would a corresponding


. However, the simple-minded predicate believespy, suitably defined, may be quite useful for expressing the facts necessary to predict someone's behavior in simpler circumstances.

Regarded as an attempt to explicate the sentence ``Ralph believes Ortcutt is a spy'', the above may be considered rather tenuous. However, we are proposing it as a notation for expressing Ralph's beliefs about Ortcutt so that correct conclusions may be drawn about Ralph's future actions. For this it seems to be adequate.

next up previous
Next: Propositions Expressing Quantification Up: FIRST ORDER THEORIES OF Previous: Replacing Modal Operators by

John McCarthy
Tue May 14 16:07:43 PDT 1996